...but with our Oyster analogues, you punch them when you enter and leave the bus so that it can charge you based on the exact amount of stops you've traversed.
Which cities do that?
Poznań. And I think that's the only one.
The system is... kinda special, as most public systems in Poland are.
Well, in Kraków you can either buy a "period" ticket: 15 minutes, 1 ride or 40 minutes, going up to 2 weeks, or you can buy a KKM (translates to Kraków City Card), which is from a month upwards, either for a specific line, or for all lines. It's fucking cheap as hell, I pay around $25 for a month for all city lines. Cards valid for suburbia are more expensive, though.
Also, the one line cards are great too, because this line constraint doesn't mean you can only ride bus/tram with a specific number, but any bus/tram as long as you're traveling the same path your line does.
There's a quite sweet city transit system in Kraków.